Calculus Examples

$lim_{x \to 0} x^{\tan (x)}$

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite $x^{\tan (x)}$ as $e^{\ln (x^{\tan (x)})}$ .

$lim_{x \to 0} e^{\ln (x^{\tan (x)})}$

Step 1.2

Expand $\ln (x^{\tan (x)})$ by moving $\tan (x)$ outside the logarithm.

$lim_{x \to 0} e^{\tan (x) \ln (x)}$

Step 2

Set up the limit as a left-sided limit.

$lim_{x \to 0^{-}} e^{\tan (x) \ln (x)}$

Step 3

Evaluate the limits by plugging in the value for the variable.

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Step 3.1

Evaluate the limit of $e^{\tan (x) \ln (x)}$ by plugging in $0$ for $x$ .

$e^{\tan (0) \ln (0)}$

Step 3.2

Since $e^{\tan (0) \ln (0)}$ is undefined, the limit does not exist.

$Does not exist$

Step 4

Set up the limit as a right-sided limit.

$lim_{x \to 0^{+}} e^{\tan (x) \ln (x)}$

Step 5

Evaluate the right-sided limit.

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Step 5.1

Move the limit into the exponent.

$e^{lim_{x \to 0^{+}} \tan (x) \ln (x)}$

Step 5.2

Rewrite $\tan (x) \ln (x)$ as $\frac{\ln (x)}{\frac{1}{\tan (x)}}$ .

$e^{lim_{x \to 0^{+}} \frac{\ln (x)}{\frac{1}{\tan (x)}}}$

Step 5.3

Apply L'Hospital's rule.

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Step 5.3.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 5.3.1.1

Take the limit of the numerator and the limit of the denominator.

$e^{\frac{lim_{x \to 0^{+}} \ln (x)}{lim_{x \to 0^{+}} \frac{1}{\tan (x)}}}$

Step 5.3.1.2

As $x$ approaches $0$ from the right side, $\ln (x)$ decreases without bound.

$e^{\frac{- \infty}{lim_{x \to 0^{+}} \frac{1}{\tan (x)}}}$

Step 5.3.1.3

Evaluate the limit of the denominator.

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Step 5.3.1.3.1

Apply trigonometric identities.

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Step 5.3.1.3.1.1

Rewrite $\tan (x)$ in terms of sines and cosines.

$e^{\frac{- \infty}{lim_{x \to 0^{+}} \frac{1}{\frac{\sin (x)}{\cos (x)}}}}$

Step 5.3.1.3.1.2

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{\cos (x)}$ .

$e^{\frac{- \infty}{lim_{x \to 0^{+}} \frac{\cos (x)}{\sin (x)}}}$

Step 5.3.1.3.1.3

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$e^{\frac{- \infty}{lim_{x \to 0^{+}} \cot (x)}}$

Step 5.3.1.3.2

As the $x$ values approach $0$ from the right, the function values increase without bound.

$e^{\frac{- \infty}{\infty}}$

Step 5.3.1.3.3

Infinity divided by infinity is undefined.

Undefined

$e^{\frac{- \infty}{\infty}}$

Step 5.3.1.4

Infinity divided by infinity is undefined.

Undefined

$e^{\frac{- \infty}{\infty}}$

Step 5.3.2

Since $\frac{- \infty}{\infty}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 0^{+}} \frac{\ln (x)}{\frac{1}{\tan (x)}} = lim_{x \to 0^{+}} \frac{\frac{d}{d x} [\ln (x)]}{\frac{d}{d x} [\frac{1}{\tan (x)}]}$

Step 5.3.3

Find the derivative of the numerator and denominator.

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Step 5.3.3.1

Differentiate the numerator and denominator.

$e^{lim_{x \to 0^{+}} \frac{\frac{d}{d x} [\ln (x)]}{\frac{d}{d x} [\frac{1}{\tan (x)}]}}$

Step 5.3.3.2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{d}{d x} [\frac{1}{\tan (x)}]}}$

Step 5.3.3.3

Rewrite $\tan (x)$ in terms of sines and cosines.

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{d}{d x} [\frac{1}{\frac{\sin (x)}{\cos (x)}}]}}$

Step 5.3.3.4

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{\cos (x)}$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{d}{d x} [1 \frac{\cos (x)}{\sin (x)}]}}$

Step 5.3.3.5

Write $1$ as a fraction with denominator $1$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{d}{d x} [\frac{1}{1} \cdot \frac{\cos (x)}{\sin (x)}]}}$

Step 5.3.3.6

Simplify.

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Step 5.3.3.6.1

Rewrite the expression.

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{d}{d x} [1 \frac{\cos (x)}{\sin (x)}]}}$

Step 5.3.3.6.2

Multiply $\frac{\cos (x)}{\sin (x)}$ by $1$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{d}{d x} [\frac{\cos (x)}{\sin (x)}]}}$

Step 5.3.3.7

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \cos (x)$ and $g (x) = \sin (x)$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{\sin (x) \frac{d}{d x} [\cos (x)] - \cos (x) \frac{d}{d x} [\sin (x)]}{\sin^{2} (x)}}}$

Step 5.3.3.8

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{\sin (x) \cdot - 1 \sin (x) - \cos (x) \frac{d}{d x} [\sin (x)]}{\sin^{2} (x)}}}$

Step 5.3.3.9

Raise $\sin (x)$ to the power of $1$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{- \sin^{1} (x) \sin (x) - \cos (x) \frac{d}{d x} [\sin (x)]}{\sin^{2} (x)}}}$

Step 5.3.3.10

Raise $\sin (x)$ to the power of $1$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{- \sin^{1} (x) \sin^{1} (x) - \cos (x) \frac{d}{d x} [\sin (x)]}{\sin^{2} (x)}}}$

Step 5.3.3.11

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{- {\sin (x)}^{1 + 1} - \cos (x) \frac{d}{d x} [\sin (x)]}{\sin^{2} (x)}}}$

Step 5.3.3.12

Add $1$ and $1$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{- \sin^{2} (x) - \cos (x) \frac{d}{d x} [\sin (x)]}{\sin^{2} (x)}}}$

Step 5.3.3.13

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{- \sin^{2} (x) - \cos (x) \cos (x)}{\sin^{2} (x)}}}$

Step 5.3.3.14

Raise $\cos (x)$ to the power of $1$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{- \sin^{2} (x) - \cos^{1} (x) \cos (x)}{\sin^{2} (x)}}}$

Step 5.3.3.15

Raise $\cos (x)$ to the power of $1$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{- \sin^{2} (x) - \cos^{1} (x) \cos^{1} (x)}{\sin^{2} (x)}}}$

Step 5.3.3.16

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{- \sin^{2} (x) - {\cos (x)}^{1 + 1}}{\sin^{2} (x)}}}$

Step 5.3.3.17

Add $1$ and $1$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{- \sin^{2} (x) - \cos^{2} (x)}{\sin^{2} (x)}}}$

Step 5.3.3.18

Simplify.

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Step 5.3.3.18.1

Simplify the numerator.

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Step 5.3.3.18.1.1

Factor $- 1$ out of $- \sin^{2} (x)$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{- (\sin^{2} (x)) - \cos^{2} (x)}{\sin^{2} (x)}}}$

Step 5.3.3.18.1.2

Factor $- 1$ out of $- \cos^{2} (x)$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{- (\sin^{2} (x)) - (\cos^{2} (x))}{\sin^{2} (x)}}}$

Step 5.3.3.18.1.3

Factor $- 1$ out of $- (\sin^{2} (x)) - (\cos^{2} (x))$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{- (\sin^{2} (x) + \cos^{2} (x))}{\sin^{2} (x)}}}$

Step 5.3.3.18.1.4

Apply pythagorean identity.

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{- 1 \cdot 1}{\sin^{2} (x)}}}$

Step 5.3.3.18.1.5

Multiply $- 1$ by $1$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{\frac{- 1}{\sin^{2} (x)}}}$

Step 5.3.3.18.2

Move the negative in front of the fraction.

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x}}{- \frac{1}{\sin^{2} (x)}}}$

Step 5.3.4

Multiply the numerator by the reciprocal of the denominator.

$e^{lim_{x \to 0^{+}} \frac{1}{x} \cdot - 1 \sin^{2} (x)}$

Step 5.3.5

Combine $\frac{1}{x}$ and $\sin^{2} (x)$ .

$e^{lim_{x \to 0^{+}} - \frac{\sin^{2} (x)}{x}}$

Step 5.4

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$e^{- lim_{x \to 0^{+}} \frac{\sin^{2} (x)}{x}}$

Step 5.5

Apply L'Hospital's rule.

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Step 5.5.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 5.5.1.1

Take the limit of the numerator and the limit of the denominator.

$e^{- \frac{lim_{x \to 0^{+}} \sin^{2} (x)}{lim_{x \to 0^{+}} x}}$

Step 5.5.1.2

Evaluate the limit of the numerator.

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Step 5.5.1.2.1

Evaluate the limit.

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Step 5.5.1.2.1.1

Move the exponent $2$ from $\sin^{2} (x)$ outside the limit using the Limits Power Rule.

$e^{- \frac{{(lim_{x \to 0^{+}} \sin (x))}^{2}}{lim_{x \to 0^{+}} x}}$

Step 5.5.1.2.1.2

Move the limit inside the trig function because sine is continuous.

$e^{- \frac{\sin^{2} (lim_{x \to 0^{+}} x)}{lim_{x \to 0^{+}} x}}$

Step 5.5.1.2.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{- \frac{\sin^{2} (0)}{lim_{x \to 0^{+}} x}}$

Step 5.5.1.2.3

Simplify the answer.

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Step 5.5.1.2.3.1

The exact value of $\sin (0)$ is $0$ .

$e^{- \frac{0^{2}}{lim_{x \to 0^{+}} x}}$

Step 5.5.1.2.3.2

Raising $0$ to any positive power yields $0$ .

$e^{- \frac{0}{lim_{x \to 0^{+}} x}}$

Step 5.5.1.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{- \frac{0}{0}}$

Step 5.5.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$e^{- \frac{0}{0}}$

Step 5.5.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 0^{+}} \frac{\sin^{2} (x)}{x} = lim_{x \to 0^{+}} \frac{\frac{d}{d x} [\sin^{2} (x)]}{\frac{d}{d x} [x]}$

Step 5.5.3

Find the derivative of the numerator and denominator.

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Step 5.5.3.1

Differentiate the numerator and denominator.

$e^{- lim_{x \to 0^{+}} \frac{\frac{d}{d x} [\sin^{2} (x)]}{\frac{d}{d x} [x]}}$

Step 5.5.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \sin (x)$ .

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Step 5.5.3.2.1

To apply the Chain Rule, set $u$ as $\sin (x)$ .

$e^{- lim_{x \to 0^{+}} \frac{\frac{d}{d u} [u^{2}] \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x]}}$

Step 5.5.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$e^{- lim_{x \to 0^{+}} \frac{2 u \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x]}}$

Step 5.5.3.2.3

Replace all occurrences of $u$ with $\sin (x)$ .

$e^{- lim_{x \to 0^{+}} \frac{2 \sin (x) \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x]}}$

Step 5.5.3.3

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$e^{- lim_{x \to 0^{+}} \frac{2 \sin (x) \cos (x)}{\frac{d}{d x} [x]}}$

Step 5.5.3.4

Simplify.

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Step 5.5.3.4.1

Reorder the factors of $2 \sin (x) \cos (x)$ .

$e^{- lim_{x \to 0^{+}} \frac{2 \cos (x) \sin (x)}{\frac{d}{d x} [x]}}$

Step 5.5.3.4.2

Reorder $2 \cos (x)$ and $\sin (x)$ .

$e^{- lim_{x \to 0^{+}} \frac{\sin (x) \cdot 2 \cos (x)}{\frac{d}{d x} [x]}}$

Step 5.5.3.4.3

Reorder $\sin (x)$ and $2$ .

$e^{- lim_{x \to 0^{+}} \frac{2 \cdot \sin (x) \cos (x)}{\frac{d}{d x} [x]}}$

Step 5.5.3.4.4

Apply the sine double-angle identity.

$e^{- lim_{x \to 0^{+}} \frac{\sin (2 x)}{\frac{d}{d x} [x]}}$

Step 5.5.3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$e^{- lim_{x \to 0^{+}} \frac{\sin (2 x)}{1}}$

Step 5.5.4

Divide $\sin (2 x)$ by $1$ .

$e^{- lim_{x \to 0^{+}} \sin (2 x)}$

Step 5.6

Evaluate the limit.

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Step 5.6.1

Move the limit inside the trig function because sine is continuous.

$e^{- \sin (lim_{x \to 0^{+}} 2 x)}$

Step 5.6.2

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$e^{- \sin (2 lim_{x \to 0^{+}} x)}$

Step 5.7

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$e^{- \sin (2 \cdot 0)}$

Step 5.8

Simplify terms.

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Step 5.8.1

Simplify the answer.

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Step 5.8.1.1

Multiply $2$ by $0$ .

$e^{- \sin (0)}$

Step 5.8.1.2

The exact value of $\sin (0)$ is $0$ .

$e^{- 0}$

Step 5.8.1.3

Multiply $- 1$ by $0$ .

$e^{0}$

Step 5.8.2

Anything raised to $0$ is $1$ .

$1$

Step 6

If either of the one-sided limits does not exist, the limit does not exist.

$Does not exist$